Found problems: 85335
(a) Several identical napkins, each in the shape of a regular hexagon, are put on a table (the napkins may overlap). Each napkin has one side which is parallel to a fixed line. Is it always possible to hammer a few nails into the table so that each napkin is nailed with exactly one nail?
(b) The same question for regular pentagons.
(A Kanel)
A family wears clothes of three colors: red,blue and green,with a separate,identical laundry bin for each color. At the beginning of the first week,all bins are empty.Each week,the family generates a total of $10 kg $ of laundry(the proportion of each color is subject to variation).The laundry is sorted by color and placed in the bins.Next,the heaviest bin(only one of them,if there are several that are heaviest)is emptied and its content swashed.What is the minimal possible storing capacity required of the laundry bins in order for them never to overflow?
Let $n$ be positive integer and $S$= {$0,1,…,n$}, Define set of point in the plane. $$A = \{(x,y) \in S \times S \mid -1 \leq x-y \leq 1 \} $$, We want to place a electricity post on a point in $A$ such that each electricity post can shine in radius 1.01 unit. Define minimum number of electricity post such that every point in $A$ is in shine area
Let $O,H$ be the circumcenter and orthocenter of acute triangle $ABC$ with $AB\neq AC$, respectively. Let $M$ be the midpoint of $BC$ and $K$ be the intersection of $AM$ and the circumcircle of $\triangle BHC$, such that $M$ lies between $A$ and $K$. Let $N$ be the intersection of $HK$ and $BC$. Show that if $\angle BAM=\angle CAN$, then $AN\perp OH$.
[b]p1.[/b] Find all triples of positive integers such that the sum of their reciprocals is equal to one.
[b]p2.[/b] Prove that $a(a + 1)(a + 2)(a + 3)$ is divisible by $24$.
[b]p3.[/b] There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle such that
(a) for each chip positioned on the circle the antipodal position is occupied by a chip of different color;
(b) there are no two neighboring blue chips.
[b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Is it possible for any finite group of people to choose a positive integer $N$ and assign a positive integer to each person in the group such that the product of two persons' number is divisible by $N$ if and only if they are friends?
* The quadratic expression $ax^2 + bx + c$ is a square (of an integer) for any integer $x$. Prove that $ax^2 + bx + c = (dx + e)^2$ for some integers d and e.
Given are $4004$ distinct points, which lie in the interior of a convex polygon of area $1$.
Prove that there exists a convex polygon of area $\frac{1}{2003}$, included in the given polygon, such that it does not contain any of the given points in its interior.
Three grasshoppers $A$, $B$ and $C$ are placed on a line. Grasshopper $B$ sits at the midpoint between $A$ and $C$. Every second, one of the grasshoppers jumps over one of the others to the symmetrical point on the other side (if $X$ jumps over $Y$ to point $X'$, then $XY $= $YX'$). After several jumps it so happened that they returned to the three initial points (but maybe in different order). Prove that in this case $B$ returns to his initial middle position.
(AK Kovaldzhy)
The Tigers beat the Sharks $2$ out of the first $3$ times they played. They then played $N$ more times, and the Sharks ended up winning at least $95\%$ of all the games played. What is the minimum possible value for $N$?
$\textbf{(A) }35\qquad\textbf{(B) }37\qquad\textbf{(C) }39\qquad\textbf{(D) }41\qquad\textbf{(E) }43$
Let $ m,n$ be two natural numbers with $ m > 1$ and $ 2^{2m \plus{} 1} \minus{} n^2\geq 0$. Prove that:
\[ 2^{2m \plus{} 1} \minus{} n^2\geq 7 .\]
Let $ABC$ be a triangle, and $I$ its incenter. Consider a circle which lies inside the circumcircle of triangle $ABC$ and touches it, and which also touches the sides $CA$ and $BC$ of triangle $ABC$ at the points $D$ and $E$, respectively. Show that the point $I$ is the midpoint of the segment $DE$.
Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes. When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospital $\textit{every}$ month, and having half the women work while only $2$ of the $7$ men work on some months means that some of the women risk getting burned out. After all, nearly all the members of the volunteer group have other jobs.
Hannah comes up with a plan that the committee likes. Beginning in October, the comittee of five volunteer teachers will consist of any five members of the volunteer group, so long as there is at least one woman and at least one man teaching each month. Under this new plan, what is the least number of months that $\textit{must}$ go by (including October when the first set of five teachers is selected, but not September) such that some five-member comittee $\textit{must have}$ taught together twice (all five members are the same during two different months)?
Do there exist distinct natural numbers $x, y, z, t$, all greater than or equal to $2$, such that $x \geq y + 2$, $z \geq t + 2$, and
\[
\binom{x}{y} = \binom{z}{t}?
\]
[i](For natural numbers $n$ and $k$ with $n \geq k$, we define $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.)[/i]
A cat can see $1$ mile in any direction. The cat walks around the $13$ mile perimeter of a triangle. Over the course of its walk, it sees every point inside of this triangle. What is the largest possible area, in square miles, of the total region it sees?
[i]2020 CCA Math Bonanza Team Round #6[/i]
Determine, with proof, the rational number $\dfrac{m}{n}$ that equals
\[\tfrac{1}{1\sqrt2+2\sqrt1}+\tfrac{1}{2\sqrt3+3\sqrt2}+\tfrac{1}{3\sqrt4+4\sqrt3}+\ldots+\tfrac{1}{4012008\sqrt{4012009}+4012009\sqrt{4012008}}\]
Find \[\int_{-4\pi\sqrt{2}}^{4\pi\sqrt{2}}\left(\dfrac{\sin x}{1+x^4}+1\right)dx.\]
Prove that $1\cdot2\cdot3\cdots 2002<\left(\frac{2003}{2}\right)^{2002}.$
Suppose triangle $ABC$ is right-angled at $B$. The altitude from $B$ intersects the side $AC$ at point $D$. If points $E$ and $F$ are the midpoints of $BD$ and $CD$, prove that $AE \perp BF$.
In triangle $ABC$, with $AB<AC$, $I$ is the incenter, $E$ is the intersection of $A$-excircle and $BC$. Point $F$ lies on the external angle bisector of $BAC$ such that $E$ and $F$ lieas on the same side of the line $AI$ and $\angle AIF=\angle AEB$. Point $Q$ lies on $BC$ such that $\angle AIQ=90$. Circle $\omega_b$ is tangent to $FQ$ and $AB$ at $B$, circle $\omega_c$ is tangent to $FQ$ and $AC$ at $C$ and both circles pass through the inside of triangle $ABC$. if $M$ is the Midpoint od the arc $BC$, which does not contain $A$, prove that $M$ lies on the radical axis of $\omega_b$ and $\omega_c$.
Proposed by Amirmahdi Mohseni
If $x^2-5x+6<0$ and $P=x^2+5x+6$ then
$\textbf{(A)}\ P \; \text{can take any real value} \qquad
\textbf{(B)}\ 20<P<30\\
\textbf{(C)}\ 0<P<20 \qquad
\textbf{(D)}\ P<0 \qquad
\textbf{(E)}\ P>30$
For all real numbers $r$, denote by $\{r\}$ the fractional part of $r$, i.e. the unique real number $s\in[0,1)$ such that $r-s$ is an integer. How many real numbers $x\in[1,2)$ satisfy the equation $\left\{x^{2018}\right\} = \left\{x^{2017}\right\}?$
Let $D$ be the foot of the altitude from $A$ in a triangle $ABC$. The angle bisector at $C$ intersects $AB$ at a point $E$. Given that $\angle CEA=\frac\pi4$, compute $\angle EDB$.
Suppose that $A$ and $B$ are sets of real numbers such that
$$A\subset B+\alpha \mathbb{Z}\quad \text{and}\quad B\subset A+\alpha\mathbb{Z}\quad \text{for all}\quad \alpha>0$$
(where $X+\alpha\mathbb=\{x+\alpha n|x\in\mathbb{X}, n\in\mathbb{Z}\}$
(a) Does it follow that $A=B$
(b) The same question, with the assumption that $B$ is bounded
$k$ is a positive integer. $A$ company has a special method to sell clocks. Every customer can reason with two customers after he has bought a clock himself $;$ it's not allowed to reason with an agreed person. These new customers can reason with other two persons and it goes like this.. If both of the customers agreed by a person could play a role (it can be directly or not) in buying clocks by at least $k$ customers, this person gets a present. Prove that, if $n$ persons have bought clocks, then at most $\frac{n}{k+2}$ presents have been accepted.